3.4 Dividing polynomials (Page 4/6)

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Section exercises

Verbal

If division of a polynomial by a binomial results in a remainder of zero, what can be conclude?

The binomial is a factor of the polynomial.

If a polynomial of degree $n$ is divided by a binomial of degree 1, what is the degree of the quotient?

Algebraic

For the following exercises, use long division to divide. Specify the quotient and the remainder.

$(x^{2} + 5 x - 1) \div (x - 1)$

$x + 6 + \frac{5}{x - 1}, quotient: x + 6, remainder: 5$

$(2 x^{2} - 9 x - 5) \div (x - 5)$

$(3 x^{2} + 23 x + 14) \div (x + 7)$

$3 x + 2, quotient: 3 x + 2, remainder: 0$

$(4 x^{2} - 10 x + 6) \div (4 x + 2)$

$(6 x^{2} - 25 x - 25) \div (6 x + 5)$

$x - 5, quotient: x - 5, remainder: 0$

$(- x^{2} - 1) \div (x + 1)$

$(2 x^{2} - 3 x + 2) \div (x + 2)$

$2 x - 7 + \frac{16}{x + 2}, quotient: 2 x - 7, remainder: 16$

$(x^{3} - 126) \div (x - 5)$

$(3 x^{2} - 5 x + 4) \div (3 x + 1)$

$x - 2 + \frac{6}{3 x + 1}, quotient: x - 2, remainder: 6$

$(x^{3} - 3 x^{2} + 5 x - 6) \div (x - 2)$

$(2 x^{3} + 3 x^{2} - 4 x + 15) \div (x + 3)$

$2 x^{2} - 3 x + 5, quotient: 2 x^{2} - 3 x + 5, remainder: 0$

For the following exercises, use synthetic division to find the quotient.

$(3 x^{3} - 2 x^{2} + x - 4) \div (x + 3)$

$(2 x^{3} - 6 x^{2} - 7 x + 6) \div (x - 4)$

$2 x^{2} + 2 x + 1 + \frac{10}{x - 4}$

$(6 x^{3} - 10 x^{2} - 7 x - 15) \div (x + 1)$

$(4 x^{3} - 12 x^{2} - 5 x - 1) \div (2 x + 1)$

$2 x^{2} - 7 x + 1 - \frac{2}{2 x + 1}$

$(9 x^{3} - 9 x^{2} + 18 x + 5) \div (3 x - 1)$

$(3 x^{3} - 2 x^{2} + x - 4) \div (x + 3)$

$3 x^{2} - 11 x + 34 - \frac{106}{x + 3}$

$(- 6 x^{3} + x^{2} - 4) \div (2 x - 3)$

$(2 x^{3} + 7 x^{2} - 13 x - 3) \div (2 x - 3)$

$x^{2} + 5 x + 1$

$(3 x^{3} - 5 x^{2} + 2 x + 3) \div (x + 2)$

$(4 x^{3} - 5 x^{2} + 13) \div (x + 4)$

$4 x^{2} - 21 x + 84 - \frac{323}{x + 4}$

$(x^{3} - 3 x + 2) \div (x + 2)$

$(x^{3} - 21 x^{2} + 147 x - 343) \div (x - 7)$

$x^{2} - 14 x + 49$

$(x^{3} - 15 x^{2} + 75 x - 125) \div (x - 5)$

$(9 x^{3} - x + 2) \div (3 x - 1)$

$3 x^{2} + x + \frac{2}{3 x - 1}$

$(6 x^{3} - x^{2} + 5 x + 2) \div (3 x + 1)$

$(x^{4} + x^{3} - 3 x^{2} - 2 x + 1) \div (x + 1)$

$x^{3} - 3 x + 1$

$(x^{4} - 3 x^{2} + 1) \div (x - 1)$

$(x^{4} + 2 x^{3} - 3 x^{2} + 2 x + 6) \div (x + 3)$

$x^{3} - x^{2} + 2$

$(x^{4} - 10 x^{3} + 37 x^{2} - 60 x + 36) \div (x - 2)$

$(x^{4} - 8 x^{3} + 24 x^{2} - 32 x + 16) \div (x - 2)$

$x^{3} - 6 x^{2} + 12 x - 8$

$(x^{4} + 5 x^{3} - 3 x^{2} - 13 x + 10) \div (x + 5)$

$(x^{4} - 12 x^{3} + 54 x^{2} - 108 x + 81) \div (x - 3)$

$x^{3} - 9 x^{2} + 27 x - 27$

$(4 x^{4} - 2 x^{3} - 4 x + 2) \div (2 x - 1)$

$(4 x^{4} + 2 x^{3} - 4 x^{2} + 2 x + 2) \div (2 x + 1)$

$2 x^{3} - 2 x + 2$

For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization.

$x - 2, 4 x^{3} - 3 x^{2} - 8 x + 4$

$x - 2, 3 x^{4} - 6 x^{3} - 5 x + 10$

Yes $(x - 2) (3 x^{3} - 5)$

$x + 3, - 4 x^{3} + 5 x^{2} + 8$

$x - 2, 4 x^{4} - 15 x^{2} - 4$

Yes $(x - 2) (4 x^{3} + 8 x^{2} + x + 2)$

$x - \frac{1}{2}, 2 x^{4} - x^{3} + 2 x - 1$

$x + \frac{1}{3}, 3 x^{4} + x^{3} - 3 x + 1$

Graphical

For the following exercises, use the graph of the third-degree polynomial and one factor to write the factored form of the polynomial suggested by the graph. The leading coefficient is one.

Factor is $x^{2} - x + 3$

Graph of a polynomial that has a x-intercept at -1.

Factor is $(x^{2} + 2 x + 4)$

$(x - 1) (x^{2} + 2 x + 4)$

Factor is $x^{2} + 2 x + 5$

Graph of a polynomial that has a x-intercept at 2.

Factor is $x^{2} + x + 1$

Graph of a polynomial that has a x-intercept at 5.

$(x - 5) (x^{2} + x + 1)$

Factor is $x^{2} + 2 x + 2$

Graph of a polynomial that has a x-intercept at -3.

For the following exercises, use synthetic division to find the quotient and remainder.

$\frac{4 x^{3} - 33}{x - 2}$

$Quotient: 4 x^{2} + 8 x + 16, remainder: - 1$

$\frac{2 x^{3} + 25}{x + 3}$

$\frac{3 x^{3} + 2 x - 5}{x - 1}$

$Quotient: 3 x^{2} + 3 x + 5, remainder: 0$

$\frac{- 4 x^{3} - x^{2} - 12}{x + 4}$

$\frac{x^{4} - 22}{x + 2}$

$Quotient: x^{3} - 2 x^{2} + 4 x - 8, remainder: - 6$

Technology

For the following exercises, use a calculator with CAS to answer the questions.

Consider $\frac{x^{k} - 1}{x - 1}$ with $k = 1, 2, 3.$ What do you expect the result to be if $k = 4 ?$

Consider $\frac{x^{k} + 1}{x + 1}$ for $k = 1, 3, 5.$ What do you expect the result to be if $k = 7 ?$

$x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$

Consider $\frac{x^{4} - k^{4}}{x - k}$ for $k = 1, 2, 3.$ What do you expect the result to be if $k = 4 ?$

Consider $\frac{x^{k}}{x + 1}$ with $k = 1, 2, 3.$ What do you expect the result to be if $k = 4 ?$

$x^{3} - x^{2} + x - 1 + \frac{1}{x + 1}$

Consider $\frac{x^{k}}{x - 1}$ with $k = 1, 2, 3.$ What do you expect the result to be if $k = 4 ?$

Extensions

For the following exercises, use synthetic division to determine the quotient involving a complex number.

$\frac{x + 1}{x - i}$

$1 + \frac{1 + i}{x - i}$

$\frac{x^{2} + 1}{x - i}$

$\frac{x + 1}{x + i}$

$1 + \frac{1 - i}{x + i}$

$\frac{x^{2} + 1}{x + i}$

$\frac{x^{3} + 1}{x - i}$

$x^{2} - i x - 1 + \frac{1 - i}{x - i}$

Real-world applications

For the following exercises, use the given length and area of a rectangle to express the width algebraically.

Length is $x + 5,$ area is $2 x^{2} + 9 x - 5.$

Length is $2 x + 5,$ area is $4 x^{3} + 10 x^{2} + 6 x + 15$

$2 x^{2} + 3$

Length is $3 x - 4,$ area is $6 x^{4} - 8 x^{3} + 9 x^{2} - 9 x - 4$

For the following exercises, use the given volume of a box and its length and width to express the height of the box algebraically.

Volume is $12 x^{3} + 20 x^{2} - 21 x - 36,$ length is $2 x + 3,$ width is $3 x - 4.$

$2 x + 3$

Volume is $18 x^{3} - 21 x^{2} - 40 x + 48,$ length is $3 x - 4,$ width is $3 x - 4.$

Volume is $10 x^{3} + 27 x^{2} + 2 x - 24,$ length is $5 x - 4,$ width is $2 x + 3.$

$x + 2$

Volume is $10 x^{3} + 30 x^{2} - 8 x - 24,$ length is $2,$ width is $x + 3.$

For the following exercises, use the given volume and radius of a cylinder to express the height of the cylinder algebraically.

Volume is $π (25 x^{3} - 65 x^{2} - 29 x - 3),$ radius is $5 x + 1.$

$x - 3$

Volume is $π (4 x^{3} + 12 x^{2} - 15 x - 50),$ radius is $2 x + 5.$

Volume is $π (3 x^{4} + 24 x^{3} + 46 x^{2} - 16 x - 32),$ radius is $x + 4.$

$3 x^{2} - 2$

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Source: OpenStax, Essential precalculus, part 1. OpenStax CNX. Aug 26, 2015 Download for free at http://legacy.cnx.org/content/col11871/1.1

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